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```190       XIII    INTEGRATION

13.9, Problems 13 and 24). Show that A(A) is either 0 or 1. (Suppose that d = A(A) e ]0,1 [.
Show that, for each component interval J of I — En, we have

A(A n J) ^ (2dl(l 4- <0)A(J).
For this purpose use (a) above and Problem l(c) of Section 13.21 to show that

dt             r1     , v      dt            rd     dt

for suitably chosen integers /?, q, r, s1 with /w — qr = ± 1. Use Problem 2 to complete
the proof.)

(c) Show that the measure p. — g • A, where g(x) = !/(!+ x), is invariant with respect
to p. Deduce from (b) and the ergodic theorem (Section 13.9, Problems 13 and 24)
that, for every A-integrable function / on I and almost all irrational x e I, we have

n~*oo n                                                                   log 2 JQ 1 -f t

(Gauss-Kuzmin formula).

(d)   Deduce from (c) that, for each integer p ^ 1, if vn(x, p) is the number of indices
k<^n such that qk(x) —p(x irrational), then

log

lOg 2

for almost all irrational x e I.
(e)   Deduce from (c) that

oo     /                 1         \ log n/log 2

lim fofrte&c) • • -^W)1'" = FT 11 + -TTV

n-»«                                      n = i \       n2 + 2n]

for almost all irrational x.

15. THE LEBESGUE-NIKODYM THEOREM AND THE  ORDER  RELATION

ON MR(X)

(13.15.1)   Let \JL and v be two positive measures on a locally compact space X,
such that v ^ ju. Then there exists a locally n-integrable function g such that

We distinguish two cases.

(I)   Suppose first that v is bounded. Since the function 1 is then v-integ-
rable, it follows from (1 3.1 1 .2.2) that for every /e Jf R(X)

(13.15.1.1) Deduce from (a) that if r=*p/q and r' — p'lq' are two rational numbers belonging
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